The Dynamics of the Forest Graph Operator

29 2 Suresh Dara, S.M. Hegde, V. Deva, S.B. Rao and T. Zaslavsky In 1966, Cummins introduced the “tree graph”: the tree graph T(G) 30 of a graph G (possibly infinite) has all its spanning trees as vertices, and 31 distinct such trees correspond to adjacent vertices if they differ in just one 32 edge, i.e., two spanning trees T1 and T2 are adjacent if T2 = T1 − e+ f for 33 some edges e ∈ T1 and f / ∈ T1. The tree graph of a connected graph need 34 not be connected. To obviate this difficulty we define the “forest graph”: 35 let G be a labeled graph of order α, finite or infinite, and let N(G) be the 36 set of all labeled maximal forests of G. The forest graph of G, denoted by 37 F(G), is the graph with vertex set N(G) in which two maximal forests F1, 38 F2 of G form an edge if and only if they differ exactly by one edge, i.e., 39 F2 = F1 − e+ f for some edges e ∈ F1 and f / ∈ F1. 40 Using the theory of cardinal numbers, Zorn’s lemma, transfinite induc41 tion, the axiom of choice and the well-ordering principle, we determine the 42 F-convergence, F-divergence, F-depth and F-stability of any graph G. In 43 particular it is shown that a graph G (finite or infinite) is F-convergent if 44 and only if G has at most one cycle of length 3. The F-stable graphs are 45 precisely K3 and K1. The F-depth of any graph G different from K3 and 46 K1 is finite. We also determine various parameters of F(G) for an infinite 47 graph G, including the number, order, size, and degree of its components. 48